Definition
- independent variable \(X\)
- dependent variable \(y\)
- analytical expression \(f(X)\)
- \[\hat{y}_{23} \approx f(X_{23})\]
Symbolic Regression Alternatives
- neural networks
- \(\frac{dy}{dX}|_{X_{23}}\approx \frac{\hat{y}_{24}-\hat{y}_{23}}{X_{24}-X_{23}}\)
- linear regression
- \(\hat{y}_{23} \approx \beta_0 +\beta_1 \cdot X\)
- best form?
- \(\hat{y}_{23} \approx \beta_2 +\beta_3 \cdot e^X\)
- \(\hat{y}_{23} \approx \beta_4 +\beta_5 \cdot log(X)\)
- \(log(\hat{y}_{23}) \approx \beta_6 +\beta_7 \cdot log(X)\)
- \(\hat{y}\approx\frac{\beta_0+\beta_1x+\beta_2x^2}{\beta_3+\beta_4x}\)
- Alternating Conditional Expectations
- Kolmogorov-Arnold Network
Typical Workflow
- what
- how
- when
- initial population
- fitness evaluation
- stop ??
- new population
Toy Dataset
\[\gamma_{API}=\frac{141.5}{\gamma_{o}}-131.5\]
| \(\gamma_{API}\) |
API Gravity |
[degAPI] |
| \(\gamma_o\) |
specific gravity |
[1/wtr] |
Software Implementation
from pysr import PySRRegressor()
myMod=PySRRegressor()
myMod.fit(x.reshape(-1, 1),y)
y_pred=myMod.predict(x.reshape(-1, 1))
Software Implementation
\(x_0-(x_0+0.013196754)+1.0131962+ \frac{x_0 (-132.5)- -141.5}{x_0}\)
\(-131.500000554+\frac{141.5}{x_0}\)
Noisy Toy
w=(random.rand(21)-0.5)*2.5
\[-132.05688+\frac{141.88547}{x_0}\]
Belly of the Beast (Encoding)
binary_operators=["+","-","*","/"]
unary_operators=None
maxsize=30
| minus |
divide |
141.5 |
specific gravity |
131.5 |
maxdepth=None
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Belly of the Beast
Stopping
niterations=100
max_evals=None
timeout_in_seconds=None
Population
populations=31
population_size=27
Quiet Toy
\[ \frac{a_{00}}{x}+a_{01}\]
w<-seq(from=121.5,to=161.5, by=10)
<expr> ::= <op>(<expr>, <expr>) | <var> | <con>
<op> ::= "+" | "-" | "*" | "/"
<var> ::= x
<con> ::= 121.5 | 131.5 | 141.5 | 151.5 | 161.5
Grammatical Evolution Search Results:
No. Generations: 1531
Best Expression: 141.5/x - 131.5
Best Cost: 0
Alternatives to Evolution
- Deterministic
- Information Technology
- Neural Networks
Alternatives to Evolution
- Deterministic
- Brute Force
- Mathematical Programming
- Sparse Regression
- Information Technology
- data-centric
- expert-centric
- Neural Networks
- Transformers
- Equation-Builders
Univariate Analysis
![]()
Raw Specific Gravity Box-and-Whiskers Plot
Bivariate Analysis
![]()
Molecular Mass vs Specific Gravity Goossens Data
Bivariate Analysis
![]()
Molecular Mass vs Boiling Point Goossens Data
Bivariate Analysis
![]()
Specific Gravity vs Boiling Point Goossens Data
Test vs. Train
![]()
Molecular Mass vs Boiling Point. Both Datasets
Test vs. Train
![]()
Molecular Mass vs Specific Gravity. Both Datasets
Existing Correlations
| \(M_w\) |
Apparent Molecular Mass |
| \(T_b\) |
True Boiling Point Temperature |
| \(\gamma_o\) |
Specific Gravity |
| \(a_{00}..a_{09}\) |
Empirical Constants |
| \(K_w\) |
Characterization Factor (intermediate value) |
| \(X_0...X_3\) |
Intermediate Variables |
Existing Correlations
Hariu & Sage (1969)
\[
M_w = a_{00} + a_{01} K_w + a_{02} K_w^2 + a_{03} T_b K_w + a_04 T_b K_w^2 + a_{05} T_b^2 K_w + a_{06} T_b^2 K_w^2
\]
\[K_w =\frac{\sqrt[3]T_b}{\gamma_o}\]
Kesler & Lee (1976)
\[M_w = X_0 + \frac{X_1}{T_b} + \frac{X_2}{T_b^2}\]
\[X_0 = a_{00} + a_{01} γ_o+ \left (a_{02} + a_{03} γ_o \right ) T_b\]
\[
X_1 = \left (1+ a_{04} γ_o + a_{05}γ_o^2 \right ) \left (a_{06} + \frac{a_ {07}}{T_b} \right ) \cdot 10^7
\]
\[
X_2 = \left (1+ a_{08} γ_o+ a_{09} γ_o^2 \right ) \left (a_{10} + \frac{a_{11}}{T_b} \right ) \cdot 10^{12}
\]
American Petroleum Institute (1977)
\[
M_w = a_{00} e^ {\left (a_{01} T_b \right )} e^{\left (a_{02} γ_o \right )} T_b^{a_{03}} γ_o^ {a_{04}}
\]
Winn, Sim & Daubert (1980)
\[M_w = a_{00} T_b^{a_ {01}} γ_o^{a_{02}}\]
Riazi & Daubert (1980)
\[M_w = a_{00} T_b^{a_ {01}}γ_o^{a_{02}}\]
Rao & Bardon (1985)
\[ln {M_w} = (a_{00} + a_{01} K_w) ln (\frac{T_b} {a_{02} + a_{03} K_w} )\]
Riazi & Daubert (1987)
\[
M_w = a_{00} T_b^{a_{01}} γ_o^{a_{02}} e^{\left (a_{03} T_b + a_{04} γ_o + a_{05} T_b γ \right )}
\]
Goossens (1996)
\[M_w = a_{00} T_b^{X_0}\]
\[
X_0 =\frac {a_{03} + a_{04} ln {\left (\frac{T_b} {a_{05} - T_b} \right )}} {a_{01} γ_o + a_{02}}
\]
Linan (2011)
\[
M_w = a_{00} e^{\left (a_{01} T_b \right )} e^{\left (a_{02} γ_o \right )} T_b^ {a_{03}} γ_o^{a_{04}}
\]
Hosseinifar & Shahverdi (2021)
\[M_w = {\left [a_{00} T_b^{a_{01}} {\left (\frac{3+2γ_o} {3-γ_o} \right )}^{\frac{a_{02}}{2}} + a_{03} T_b^{a_{04}} {\left (\frac{3+2γ_o}{3-γ_o} \right )}^{\frac{a_{05}}{2}} \right ]}^{a_{06}}\]
Stratiev (2023)
\[
M_w = a_{00} + a_{01} e^{\left [a_{02} e^{\left (a_{03} \frac{T_b^{a_{06}}}{γ_o^{a_{05}}} \right )} \right ]}
\]
Operators to Consider
Partial List
| pow |
binary |
one expression raised to the power of another |
| log |
unary |
logarithm of an expression |
| exp |
unary |
antilogarithm of an expression |
| sqr |
unary |
expression squared |
| cub |
unary |
expression cubed |
| inv |
unary |
inverse of an expression |
Power Run
\[M_w=a_{00} \cdot a_{01}^{\gamma_o^{-a_{02}} + a_{03} \cdot T_b} + a_{04}\]
Correlation Coefficients
| Raw MW |
1.000000 |
0.997281 |
| This Equation |
0.997281 |
1.000000 |
Power Run
![]()
Exponential Run
\[ M_w= - T_b \cdot \left(a_{00} \cdot T_b - a_{01}\right) \left(a_{02} \cdot 10^{-6} \left(a_{03} \cdot \gamma_o - 2 \cdot T_b \right) \left(T_b - a_{04}\right) - 1\right) + a_{05}\]
\[ M_w= a_{00}\cdot T_b + a_{01} \cdot e^{- \gamma_o^{2} + a_{02}\cdot \gamma_o + a_{03} \cdot T_b}\]
Correlation Coefficients
| Raw MW |
1.000000 |
0.997705 |
0.998420 |
| First Equation |
0.997705 |
1.000000 |
0.999497 |
| Second Equation |
0.998420 |
0.999497 |
1.000000 |
![]()
Raw Data Scale Change
First Sparse Run
\[ M_w=-a_{00} \cdot \gamma_o +\frac{a_{01}\cdot \gamma_o}{T_b}+a_{02}\cdot T_b -a_{03}\]
Correlation Coefficients
| Raw Mw |
1.000000 |
0.970451 |
| This Equation |
0.970451 |
1.000000 |
First Sparse Run
![]()
Second Sparse Run
\[ M_w=-a_{00}\cdot \gamma_o\cdot T_b +a_{01}\cdot T^2_b +a_{02}\cdot e^{\gamma_o} +a_{03}\]
Correlation Coefficients
| Raw Mw |
1.000000 |
|
| 1st Run Equation |
0.970451 |
|
| 2nd Run Equation |
0.985800 |
|
Second Sparse Run
![]()
Validation Runs
| Raw Mw |
1.000000 |
| Default |
0.999957 |
| Goossens Correlation |
0.999939 |
| Power |
0.996964 |
| Exponential #2 |
0.998954 |
| Aeon #1 |
0.999973 |
| Aeon #2 |
0.999921 |
| Box-Cox |
0.999696 |
| Sparse #1 |
0.992303 |
| Sparse #2 |
0.996331 |
General Observations
- UOP Characterization Factor
- Replication
- Explainability
- Table Look-Up
\[ M_w^t = \left (\gamma_o \left(a_{00} T_b -a_{01} \right) log{(\gamma_o)} + a_{02} \right) log{(T_b)} \]
